In Figure REF , we provide a graphical illustration of procedures considered in the nonlinear approach that we propose here. Again, as steps 1 and 2 of our data-driven framework shown in Figure REF  are similar for both the linear and nonlinear approaches, we here begin our procedure at the third step. The main difference between the linear and nonlinear approaches lie within the compression as well as the reconstruction tool. For linear compression we use POD and for the reconstruction we use the \(L_2\)  projection. For the nonlinear compression (of \(T_h \left(\mathbf {\mu }^{ \left(1\right)}\right) , \cdots , T_h \left(\mathbf {\mu }^{\left(\mathrm {M}\right)}\right)\) ) and reconstruction we utilize AE. More specifically, for the compression we utilize the encoder, whereas for the reconstruction we utilize the decoder of the AE. To reiterate, we note that \(T_h \left(\mathbf {\mu }^{ \left(i\right)}\right)\)  represents \(T_h\)  given \(\mathbf {\mu }^{\left(i\right)}\) , belonging to the training set, evaluated at all space and time. The AE could be considered as a nonlinear variation of POD [1]}, [2]}, [3]}, and it uses the so-called bottleneck inside its hidden layers to force the networks to identify a nonlinear manifolds (\({z}_1^T, \cdots , {z}_Q^T\) ) of the training examples [2]}, [5]}.
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